I am analyzing an categorial unbalanced panel data set of the following structure:

FirmID Year Region Sector ROE PB

The data set consist of annual data for about 1’500 firms covering 25 years. Each firm is assigned to one of 5 regions and to one of 11 sectors. PB is the dependent variable, ROE the independent variable.
The aim of the study is to investigate the linear relationship (if any) between PB and ROE by OLS and by Panel Data regression for different sub-sets of the original sample, eg. for particular Regions or Sectors and/or various combinations thereof. The strength of the relationship will be measured by the R2 of the regression.

It is my current understanding that a fixed effects panel regression model is the most appropriate way of analyzing such data and hence the Stata code would look as follows

For OLS regression:

regress PB ROE if Region ==1

For FE Panel data regression:

xtreg PB ROE if Region ==1, fe

I wonder if anyone with expertise in panel data analysis could give some feedback or advice whether the above approach is indeed correct?

Suppose the following two (hypothetical) scenarios:
  1. For Region ==1, PB and ROE show no linear relation (low R2) in any particular year (looking at all firms), however there is a strong serial correlation over time for each individual firm.
  2. For Region ==1, PB and ROE show a linear relation (high R2) in any particular year, however there is no linear relation over time for each individual firm.
How would these two scenarios be reflected in the R2 of the panel data regression and is the fixed-effects model as specified above indeed the best model for both scenarios? If not, which other model would be better to use and how to decide for it?

Thank you very much in advance for any feedback and hints!